Dear Reader,
Below I have extracted, for your reading pleasure and intellectual enjoyment, the section from pages II-11 through II-14 of F.E.D. Vignette #4, The Goedelian Dialectic of the Standard Arithmetics.
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The section quoted below is exemplary as to how other dialectical-mathematical 'meta-models', formulated using the NQ_ dialectical algebra, may be translated into the format of a simple Question/Answer dialogue, based upon concepts sourced from the tradition of "Systematic Dialectics" methods of presentation.
"In resonance with the prologue to the quote from Nicholas Rescher extracted in section B.gamma. of Part I.: “Already the Socrates of Plato’s Theaetetus conceived of inquiring thought as a discussion or dialogue that one carries on with oneself.” [ibid., p. 46], the ‘Dyadic Seldon Function ‘meta-model’ central to this essay can be most directly interpreted -- especially taking into account what we have had to say above regarding ‘‘‘personification’’’ and '''impersonation'''-- as ‘meta-modeling’ a generic such ‘self-dialogue’, as a readily reusable ‘recipe for thought’ for a systematic and well-ordered refresher for one’s self, and for re-presentation to others, of the contemporary “Standard Arithmetics”.
However, in Part I., version 2, we translated our eight-category Dyadic Seldon Function model for the systematic dialectic of the Value-Form content in Marx’s Capital into the narrative format of an ‘exo-allo-dialogue’ between two generic, fictive interlocutors, rather than into an ‘endo-auto-dialogue’ within the mind of a single such generic, fictional “character”.
We can do likewise with our Dyadic Seldon Function ‘meta-model’ presentation for the ‘meta-system’ of the “Standard Arithmetics”, translating it into a somewhat Socratic, ‘dialogue-ic’ “Q&A”, in a way which perhaps more compellingly reveals what is implicitly involved in the Seldon-Function categorial progression algorithm, as applied to this subject-matter.
A sample of such a translation is exhibited below.
In reviewing the dialogue below, it is important to keep in mind that the ‘‘‘analysis of categories’’’, referenced repeatedly therein, has a special meaning in this context.
The ‘‘‘analysis of a category’’’ means, herein, the process of elaboration of the content of that category, in the sense of the ‘explicitization’ of at least some of its content which would otherwise remain implicit, i.e., if only that category’s name were given: the explicit evocation of some, at least, of the sub-categories of that category, and, perhaps, of some of their sub-categories as well, i.e., the elaboration of at least some of the category’s ‘sub-sub-categories’, or ‘sub^2-categories’, as well.
‘‘‘Analysis’’ in this, dialectical, sense, is the process of human cognition by which we “ascend” [Marx, reversing dialectical predecessors Plato, et al.] to greater detail, or to greater “determinateness” and ‘thought-concreteness’, from greater abstraction and abstract simplicity.
It is the oppositely-directed cognitive movement/movement-of-cognition to that of ‘‘‘Synthesis’’’, whereby we “descend” from multiple, more detailed, more specific ‘sub^n-categories’, to a single, more general, more generic, more “abstract” and simplified ‘sub^(n-1)-category’, one that embraces, and ‘[re-]implicitizes’ into itself, all of the ‘sub^n-categories’ immediately ‘“above”’ it, as well as all of those ‘sub^(n+1)+-categories’ ‘“above”’ them [if any].
Q1: ¿What is the simplest category that grasps the totality of our -- and, in our view, of most people’s -- ‘untheorized’, ‘unsystematic’, “chaotic” [Marx] experience, and knowledge, of the modern / contemporary Standard Arithmetics?
A1: The category of the “Natural” system of arithmetic, the system which is about the
N = { I, II, III, ... }
set of numbers, and whose analysis is as follows: ... . It is the system that ‘explicitizes’ "Standard Numbers" as “counts” -- as cardinal numbers.
Q2: ¿Does the analysis of this arithmetic-system category, N#, exhaustively systematize our previously fragmentary experience and “chaotic” knowledge of the totality of Standard Arithmetic, accounting entirely for/explaining all of the ‘ideo-phenomena’ of number that we encounter today? ¿Or, are there other categories needed to entirely comprehend that experience and knowledge, to fully classify/systematize and explain those ‘ideo-phenomena’ of number; categories whose content is not explicitly covered by the analysis of the “Natural Arithmetic” system-category; categories whose analysis could therefore add to our systematic grasp of the totality of the systems of modern/contemporary “Standard Arithmetic”? We offer into evidence here the N-algebraic equation
n + x1 = n, for all n in N,
not satisfiable by any number in N: x1 is not an element of N.
And this equation is paradoxical from the point of view of the N# definition of ‘Standard Number’, of number “Natural-ness”, for which addition cannot result in no increase in numeric magnitude.
A2: The category of the arithmetical sub-system of the ‘‘‘zeros’’’, or of the ‘aught numbers’ --
a = { I-I = 0, II-II = 0, III-III = 0, ... } --
the sub-system category denoted by a# -- is needed to satisfy the kind of equation that you cited. Your equation transforms, algebraically, to --
x1 = n - n,
-- and so invokes the set of results of ‘self-subtraction’ for all “Naturals”, n. The analysis of the category of ‘the aughts’ ‘explicitizes’ an expansion of the meaning/definition of ‘Standard Number’, over and above that explicit in the ‘N-numbers’ concept: . . ..
The upshot of this analysis is that not just “counts”, but also ‘no[n]-counts’ -- certain kinds of ‘not-counts’ -- are also among the ‘Standard Numbers’.
Q3: ¿Do the [sub-]system categories of the “Naturals” and the ‘aughts’, of “counts” as ‘Standard Numbers’ versus of ‘no[n]-counts’ as ‘Standard Numbers’, simply represent two irreconcilable, absolutely disparate and separated and, in some ways, diametrically opposite categories? ¿Or, do we find, in our experience of the human number meme, that these two categories interrelate, and even combine? ¿That is, does the ‘no[n]-counts’ category, the counter-example to, and the breakdown of, the “counts” concept/definition of ‘Standard Number’, unite with the category that it counters, to form a new whole system-of-arithmetic category? ¿Can our explanation/systematization/classification of the ‘meta-system’ of the “Standard Arithmetics” be adequate as either the analysis of the category N# by itself, or the analysis of category a# by itself, as two, distinct, disjunct alternatives?
A3: “Yes”, the “Naturals” and the ‘aughts’ are two different, separate, disparate, even qualitatively opposite categories of kinds of number, of number ‘ideo-ontology’. But “No”, they are not absolutely separate, or absolutely separable, in our experience of humanity’s numbers meme. And our theory/presentation of the “Standard Arithmetics” ‘meta-system’ is inadequate to that experience if it takes a# and N# as only dirempt. Indeed, in our experience/knowledge of modern/contemporary Arithmetics, they both are ingredient in, and inseparable as, the vastly superior numeration system -- vastly superior to the Roman Numerals and to the other ancient, additive/'juxtapositional' numeration-systems -- that is the Indo-Arabic numeration system, the "place-value" numeration system made possible by the advent of 0 as "place-holder", and as full number in its own right, in which separate a# and N# give way to their ‘“complex unity”’, to the W# system which the former form by their ‘“combination”’. The ‘‘‘combination’’’ of the “Naturals” and the ‘aughts’ constitutes the “Wholes” -- the “Whole numbers”, such that
W = { 0...0, 0...01, 0...02, 0...03, ... },
and in which the equation
w + x1 = w, for all w in W,
which is a ‘‘‘well-formed equation’’’ in W#, can “now” be satisfied, namely, by
x1 = 0, x1 is an element of W.
The analysis of the W# category yields a concept/definition of ‘Standard Number’ that encompasses and reconciles the “counts” & ‘no[n]-counts’ definitions of number, in a single concept/definition of number ‘‘‘Whole-ness’’’.
Q4: ¿Does the analysis of this new, composite, successor system-of-Standard-Arithmetic category, W#, exhaust our experience and knowledge of the totality of contemporary/modern Standard Arithmetic(s)? ¿Does that analysis account entirely for, i.e., does it explain, that experience and knowledge? ¿Or are there other categories that are required to cover, and to systematize, that experience and knowledge; required to classify, and to explain, all of it -- or, at least, more of it: including even [at least some of] the most exotic ‘ideo-phenomena’ of modern/contemporary Standard Arithmetic(s) that we have encountered; other categories whose content is not explicitly elaborated by the analysis of the “Whole”-numbers arithmetic-system category; other categories whose analysis/‘content-explicitization’ would therefore add to our comprehension of the totality of modern/contemporary Standard Arithmetic(s)? I call your attention, specifically, to the algebraic equation
w + x2 = 0,
0 & w in W, which is a ‘‘‘well-formed equation’’’ in W# [i.e., ‘w’, ‘+’,‘x’, ‘2’ [including as a subscript], ‘=’, & ‘0’ are all parts of the “official vocabulary” of W#]. Despite this equation’s being an equation of W# -- a ‘‘‘well-formed’’’, “legal” equation in W# [though not so in N#, because 0 is not in N] -- this equation is not satisfiable by any number in W: x2 is not in W. This equation defines an ‘ideo-phenomenon’ of ‘decreasive addition’, one which is paradoxical from the point of view of the new, W#, definition of ‘Standard Number’, because the possibility of ‘decreasive addition’ is not instantiated within W#. The nature of number as implied and specified in W# is contradicted by the “number” whose existence is implied by this equation, under the name of its “unknown”, x2. The behavior ascribed, by this equation as a whole, to the number, x2, is of a kind-of-number(s) which, when added to any W-number, “nullifies” it, decreasing it all the way back to 0.
A4: The analysis of the W# composite category as a whole, and of each of its constituent sub-categories -- N#, a#, & #qaN -- which, taken together, constitute/connote W#, analyzed again, individually, but now in the context of the analysis of W# as a whole, does further advance our detailed comprehension of our experience and knowledge of the modern/contemporary Standard Arithmetic(s), #. But, all of that analysis still leaves out[side of itself] fundamental aspects of our experience and knowledge of the ‘ideo-phenomena’ of this ‘human-phenomic’ totality that we ‘‘‘name’’’ #.
In particular, the category of the axiomatic system-component for the “minus numbers”, m#, is needed to solve the equation
w + x2 = 0, in which x2 functions as an “additive inverse” of w.
The analysis of this “minus numbers” category is needed to advance our comprehension of the modern/contemporary Standard Arithmetic(s) beyond the ken of the W# category. The analysis of the axiomatic sub-system category of the “minus numbers”, or of the “negative[ly]-signed” numbers, such that
m = { ±0-I = -1, ±0-II= -2, ... },
'explicitizes' the following expansion of the meaning -- of the definition -- of the ‘Standard Numbers’: . . ..
The upshot of this expansion is that numbers, and that counting, and that ‘‘‘counting numbers’’’, can have direction-- can have one of two ‘‘‘co-linear directionalities’’’, in terms of the number-“line” convention of the ‘‘‘analytic-geometric’’’ visualization of the ‘‘‘number-spaces’’’ of these number-systems: an explicit right-hand directionality [labeled by the ‘+’ sign], or an explicit left-hand directionality [labeled by the ‘-‘ sign], or, in just a "single" case -- the case of ±0 -- directional ‘‘‘neutrality’’’/‘intra-bi-ality’ [‘±’].
Q5: ¿Do the [sub-[axioms-]]systems categories of the arithmetic of the “Wholes” & of the arithmetic of the ‘minuses’, of the apparently/superficially ‘undirectional’, “direction-less”, or “mono-directional counts” as ‘Standard Numbers’ versus of explicitly ‘other-directional counts’ as ‘Standard Numbers’, simply form two irreconcilable, absolutely disparate and separated and, in some ways, diametrically opposite categories? ¿Or, do we find, in our experience of the human number meme, that these two categories interrelate, and even combine? ¿That is, does the ‘other-directional counting’ category, m#, the counter-example to, and ‘demotor’ of, the W# concept/definition of ‘Standard Number’, which signals the failure of the W# category to constitute the totality of Standard Arithmetic, unite with the category that it counters, demotes, and criticizes, to form an integral new system-of-arithmetic category? ¿Can our sought-after total explanation/systematization/classification of the ‘meta-system’ of the “Standard Arithmetics” be adequate as either the analysis of the category W# by itself, or the analysis of category m# by itself, seen as radically disjunct alternatives?
A5: “Yes”, the “Wholes” and the ‘minuses’ are two different, separate, disparate, even qualitatively opposite categories of kinds of number, of number ‘ideo-ontology’. But “No”, they are not absolutely separate, nor absolutely separable, in our experience and knowledge of humanity’s numbers meme. And our theory/presentation of the “Standard Arithmetics” ‘meta-system’ is inadequate to that experience if it takes W# and m# only as dirempt. Indeed, we have encountered an axiomatic system-of-arithmetic category in which, and by which, the three -- the explicitly-[minus-]signed “minus numbers”, the explicitly-[plus-]signed “plus-numbers”, and, at their meeting point, between them, the explicitly-[double-]signed ‘neutral number’ [±0] -- are all integrated into a single system of arithmetic, the system of the integers, Z#. Per this system-category, Z#, we have
Z = { ..., -3, -2, -1, ±0, +1, +2, +3, ... }.
The explicit three-fold bi-directional classification of numbers is effected in Z#. That is, each number in Z has either a leftward [-] direction, or a rightward [+] direction, or no direction at all/neither direction, or both directions at once, with equal “weight” [±]. If a pair of ‘oppositely-directioned’ Z#-numbers that are also of equal magnitude [of equal “absolute value”] combine additively, they ‘‘‘mutually annihilate’’’ undoing both their directions and their magnitudes, yielding/leaving/arriving at ±0:
w + x2 = ±0, if x2 = -w,
and although it is true that -w is NOT in W, it is also true that -w IS in m, which, in turn, is a subset of Z.
Thus, in this m#-and-W#-synthesizing axioms-system, the equation
w + x2 = ±0 can be satisfied.
The composite axioms-system category Z# ‘‘‘contains’’’ seven axioms sub-systems sub-categories: (a.) the 3 past-stages categories N#,a#,& #qaN, which, added together, = W#, which are/is ‘evolutely’ conserved in Z#, plus (b.) m#, the core of Z#, plus; (c.) #qmN, connoting the conversion of the N# into a part of the Z#, plus; (d.) #qma, connoting conversion of a# into another part of Z#, plus; (e.) #qmaN, connoting ‘m#-ization’ of #qaN. We have thusly found our way to a ‘‘‘meta-system transition’’’ [cf. Turchin],
W# ---) Z#,
a transition within the ‘‘‘meta-system’’’ which is formed by the totality of these Gödelian-Dialectical transitions --
N# ---) W# ---) Z# ---) Q# ---) R# ---) C# ---) H# ---) . . .
-- just as, earlier in this dialogue, we found our way similarly to the ‘‘‘meta-system transition’’’
N# ---) W#.
Q6: ¿Does the analysis of this new, composite, successor system-of-Standard-Arithmetic category, Z#, exhaustively systematize our experience and knowledge of the totality of contemporary/modern Standard Arithmetic(s)? ¿Does that analysis account entirely for, i.e., does it explain, that experience and knowledge? ¿Or are there other categories, already encountered by us unsystematically, in our “chaotic” [Marx] experience and fragmentary knowledge of modern/contemporary Standard Arithmetics, that are required to more fully cover, and to more fully systematize, and to render more fully intelligible, that experience and knowledge; required to classify, and to explain, all of it -- or, at least, more of it: even [at least some of] the most exotic ‘ideo-phenomena’ of modern/contemporary Standard Arithmetic(s) that we have ever encountered? ¿Are there other categories whose content is not explicitly elaborated by the analysis of the “Integers” arithmetic-system category, Z#, and of its sub-categories, in the context of the “Integers” system-category as a whole? ¿Are there other categories whose analysis/‘content-explicitization’/elaboration would therefore add to/improve/increase the coverage of our theory’s comprehension of [another part of] the totality of the ‘ideo-phenomena’ of modern/contemporary Standard Arithmetic(s)? . . . . . . . . . . . . . . . "
Regards,
Miguel
